with the original recurrence divided into $k$ separate solutions each defined by initial conditions $f_{j,k}(x)$ and $f_{j+k,k}(x)$, $0\leq j < k$, where the functions $a_k(x)$ and $b_k(x)$ are themselves defined by a specific pair of recurrence relations in $k$. But the modular structure's giving me troubles in proving either the collapse or, given the collapse, the validity of the recurrences potentially defining $a_k(x)$ and $b_k(x)$ through any sort of inductive approach, and I haven't had any luck digging up any references to a recurrence structure of even a vaguely similar form. Even just something similar in nature could give me a lead into pinning this down.

I have no idea, but at least I can verify a tiny bit. If $u_n$ is given by $au_{n-1}+bu_{n-2}$, $u_{n-1}+abu_{n-2}$, or $u_{n-1}+bu_{n-2}$ according as $n$ is 1, 2, or 0 modulo 3, then (at least for $n\equiv1\pmod3$) we get $u_n=(2ab+a+b)u_{n-3}+ab^3u_{n-6}$, which agrees with your suggested form. But perhaps this was already found in your "preliminary work".
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Gerry MyersonDec 6 '11 at 21:56

It was, yes, though I appreciate it anyway; so far I've verified it by hand for $k$ from 2 to 5.
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Justin HilyardDec 7 '11 at 4:58